Low-lying collective excited states in nonintegrable pairing models based on the stationary-phase approximation to the path integral

For a description of large-amplitude collective motion associated with nuclear pairing, requantization of time-dependent mean-field dynamics is performed using the stationary-phase approximation (SPA) to the path integral. We overcome the difficulty of the SPA, which is known to be applicable to integrable systems only, by developing a requantization approach combining the SPA with the adiabatic self-consistent collective coordinate method (ASCC+SPA). We apply the ASCC+SPA to multilevel pairing models, which are nonintegrable systems, to study the nuclear pairing dynamics. The ASCC+SPA gives a reasonable description of low-lying excited $0^{+}$ states in nonintegrable pairing systems.

Figure 1

Classical trajectories satisfying the EBK quantization condition Eq. (17) with $k=1$ and 2 in the $(\varphi ,j)$ phase space. The crosses, solid and dashed lines correspond to the results of the ASCC+SPA, TDHFB2+SPA, and TDHFB+SPA, respectively. The crosses for the ASCC+SPA trajectories are plotted every ten steps of the time evolution ($\delta q=10dq=0.1/\sqrt{{\epsilon }_0}$).

Figure 2

Calculated action integrals for the $|0_2^{+}\rangle$ and $|0_3^{+}\rangle$ states as functions of time $t$. The crosses, solid and dashed lines correspond to the ASCC+SPA, the TDHFB2+SPA, and the TDHFB+SPA, respectively. The action integrals are calculated on each trajectory in Fig. 1 from $(\varphi ,j)=(0,j_{\max })$ in the clockwise direction. The crosses for the ASCC+SPA trajectories are plotted every ten steps of the time evolution ($\delta q=10dq=0.1/\sqrt{{\epsilon }_0}$).

Figure 3

Occupation probabilities for the $0_2^{+}$ and $0_3^{+}$ states. The horizontal line indicates the $j=(m_2-m_1)/2$ of the quasispin basis in Eq. (44). The vertical bars at each $m_2-m_1$ from left to right represent $|C_m{|}^2$ of Eq. (45) in the ASCC+SPA, TDHFB2+SPA, and TDHFB+SPA calculations, respectively.

Figure 4

Eigenvalues of moving-frame QRPA equation as a function of the collective coordinate $q^1$, from $N=14$ to $N=24$. The thick (green) lines are the modes we choose as the collective coordinate $q^1$, while the dashed lines correspond to the zero modes ($q^n$). In each panel, both ends of the horizontal axis corresponds to the ending points of the collective path $q^1$.

Figure 5

Occupation numbers $2j^{\alpha }$ in each single-particle level as a function of the collective coordinate $q^1$, from $N=14$ to $N=24$. The purple, green, and blue lines correspond to $\alpha =1$, 2, and 3, respectively. At the left end point of the collective coordinate in each panel, the configuration corresponds to the “HF-like” states. See text for details.

Figure 6

Collective potential $V\left(q^1\right)$ obtained from the ASCC. We adjust the energy minimum point as $q^1=0$ and $V=0$.

Figure 7

Calculated strength of pair-addition transition Eq. (48) from $N=14$ to $N=22$. The solid (dashed) lines correspond to the ASCC+SPA (exact) calculation. The horizontal line indicates the particle number of the initial states. The upper panel shows the intraband transitions, $0_1^{+}\to 0_1^{+}$ and $0_2^{+}\to 0_2^{+}$, while the lower panel shows the interband transitions, $0_1^{+}\to 0_2^{+}$ and $0_2^{+}\to 0_1^{+}$.

Figure 8

The same as described in the caption of Fig. 4 but for Pb isotopes.

Figure 9

The same as described in the caption of Fig. 5 but for Pb isotopes.

Figure 10

The same as described in the caption of Fig. 6 but for Pb isotopes.

Figure 11

The same as described in the caption of Fig. 7 but for Pb isotopes.

Figure 12

Pairing gap as a function of collective coordinate $q^1$ in ${}^{192}\mathrm{Pb}$.